Theorem kur14lem9 32463

Description: Lemma for kur14 32465. Since the set 𝑇 is closed under closure and complement, it contains the minimal set 𝑆 as a subset, so 𝑆 also has at most 14 elements. (Indeed 𝑆 = 𝑇, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
kur14lem.j	⊢ 𝐽 ∈ Top
kur14lem.x	⊢ 𝑋 = ∪ 𝐽
kur14lem.k	⊢ 𝐾 = (cls‘𝐽)
kur14lem.i	⊢ 𝐼 = (int‘𝐽)
kur14lem.a	⊢ 𝐴 ⊆ 𝑋
kur14lem.b	⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))
kur14lem.c	⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴))
kur14lem.d	⊢ 𝐷 = (𝐼‘(𝐾‘𝐴))
kur14lem.t	⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))
kur14lem.s	⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}

Assertion

Ref	Expression
kur14lem9	⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐾   𝑥,𝑦,𝑇   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑦)

Proof of Theorem kur14lem9

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kur14lem.s	. . 3 ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}
2		vex 3499	. . . . . 6 ⊢ 𝑠 ∈ V
3	2	elintrab 4890	. . . . 5 ⊢ (𝑠 ∈ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ↔ ∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥))
4		ssun1 4150	. . . . . . . 8 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
5		ssun1 4150	. . . . . . . . 9 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ⊆ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))})
6		ssun1 4150	. . . . . . . . . 10 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ⊆ ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))
7		kur14lem.t	. . . . . . . . . 10 ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))
8	6, 7	sseqtrri 4006	. . . . . . . . 9 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ⊆ 𝑇
9	5, 8	sstri 3978	. . . . . . . 8 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ⊆ 𝑇
10	4, 9	sstri 3978	. . . . . . 7 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇
11		kur14lem.j	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12		kur14lem.x	. . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽
13	12	topopn 21516	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
14	11, 13	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑋 ∈ 𝐽
15	14	elexi 3515	. . . . . . . . 9 ⊢ 𝑋 ∈ V
16		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
17	15, 16	ssexi 5228	. . . . . . . 8 ⊢ 𝐴 ∈ V
18	17	tpid1 4706	. . . . . . 7 ⊢ 𝐴 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}
19	10, 18	sselii 3966	. . . . . 6 ⊢ 𝐴 ∈ 𝑇
20		kur14lem.k	. . . . . . . . 9 ⊢ 𝐾 = (cls‘𝐽)
21		kur14lem.i	. . . . . . . . 9 ⊢ 𝐼 = (int‘𝐽)
22		kur14lem.b	. . . . . . . . 9 ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))
23		kur14lem.c	. . . . . . . . 9 ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴))
24		kur14lem.d	. . . . . . . . 9 ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴))
25	11, 12, 20, 21, 16, 22, 23, 24, 7	kur14lem7 32461	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑇 → (𝑦 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
26	25	simprd 498	. . . . . . 7 ⊢ (𝑦 ∈ 𝑇 → {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇)
27	26	rgen 3150	. . . . . 6 ⊢ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇
28	25	simpld 497	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ⊆ 𝑋)
29	15	elpw2 5250	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
30	28, 29	sylibr 236	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ∈ 𝒫 𝑋)
31	30	ssriv 3973	. . . . . . . 8 ⊢ 𝑇 ⊆ 𝒫 𝑋
32	15	pwex 5283	. . . . . . . . 9 ⊢ 𝒫 𝑋 ∈ V
33	32	elpw2 5250	. . . . . . . 8 ⊢ (𝑇 ∈ 𝒫 𝒫 𝑋 ↔ 𝑇 ⊆ 𝒫 𝑋)
34	31, 33	mpbir 233	. . . . . . 7 ⊢ 𝑇 ∈ 𝒫 𝒫 𝑋
35		eleq2 2903	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑇))
36		sseq2 3995	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → ({(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
37	36	raleqbi1dv 3405	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
38	35, 37	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → ((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) ↔ (𝐴 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇)))
39		eleq2 2903	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → (𝑠 ∈ 𝑥 ↔ 𝑠 ∈ 𝑇))
40	38, 39	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) ↔ ((𝐴 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇) → 𝑠 ∈ 𝑇)))
41	40	rspccv 3622	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) → (𝑇 ∈ 𝒫 𝒫 𝑋 → ((𝐴 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇) → 𝑠 ∈ 𝑇)))
42	34, 41	mpi 20	. . . . . 6 ⊢ (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) → ((𝐴 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇) → 𝑠 ∈ 𝑇))
43	19, 27, 42	mp2ani 696	. . . . 5 ⊢ (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) → 𝑠 ∈ 𝑇)
44	3, 43	sylbi 219	. . . 4 ⊢ (𝑠 ∈ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} → 𝑠 ∈ 𝑇)
45	44	ssriv 3973	. . 3 ⊢ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ⊆ 𝑇
46	1, 45	eqsstri 4003	. 2 ⊢ 𝑆 ⊆ 𝑇
47	11, 12, 20, 21, 16, 22, 23, 24, 7	kur14lem8 32462	. 2 ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)
48		1nn0 11916	. . 3 ⊢ 1 ∈ ℕ0
49		4nn0 11919	. . 3 ⊢ 4 ∈ ℕ0
50	48, 49	deccl 12116	. 2 ⊢ ;14 ∈ ℕ0
51	46, 47, 50	hashsslei 13790	1 ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {crab 3144   ∖ cdif 3935   ∪ cun 3936   ⊆ wss 3938  𝒫 cpw 4541  {cpr 4571  {ctp 4573  ∪ cuni 4840  ∩ cint 4878   class class class wbr 5068  ‘cfv 6357  Fincfn 8511  1c1 10540   ≤ cle 10678  4c4 11697  ;cdc 12101  ♯chash 13693  Topctop 21503  intcnt 21627  clsccl 21628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-dju 9332  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-xnn0 11971  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-hash 13694  df-top 21504  df-cld 21629  df-ntr 21630  df-cls 21631

This theorem is referenced by:  kur14lem10  32464